Solve by Completing the square
Completing the square "change" the equation from
ax2 + bx + c = 0 => (x + d)2 = e. where d and e are numerical.
ax2 + bx + c = 0
Divide the equation by a
=> x2 + bx/a + c/a = 0
Applying (x + ky)2 = (x +k/2)2 - (K/2)2
(x + b/2a)2 - (b/2)2 + c/a = 0
(x+ b/2a)2 = (b/2)2 - c/a
Solving the equation for x:
x + b/2a = + √ (b/2)2 - c/a
x = -b/2a + √ (b/2)2 - c/a
x = -b/2a + √ (b/2)2 - c/a
Or x = -b/2a - √ (b/2)2 - c/a
Formula : ax2+ bx + c = a(x+ b/a)2– (b/a)2+ c/a = 0
(x+ b/2a)2 = (b/2a)2 - c/a
=> x = -b/2a + √ (b/2)2 - c/a Or
x = -b/2a - √ (b/2)2 - c/a
Example
Solve 2x2+ 4x + 1 = 0
Using completing the square ax2+ bx + c = a(x+ b/a)2– (b/a)2+ c
2x2 + 6x + 1 = 2(x2 + 4x + 1/2) = 0
x2 + 4x + 1/2 = 0
(x + 4/2)2– (4/2)2 + 1/2 = 0
(x + 2)2 = (2)2 - 1/2
(x + 2)2 = 4 – 1/2
(x + 2)2 = 7/2
x + 2 = +√ 7/2
x = -2+√ 7/2
x = -2 + √ 7/2 or x = -2 - √ 7/2
Graphical Method
Using the above equation x2 + 4x + 1 = 0 to plot a graph,
(x + 2)2 = 7/2
=> The minimum point is (-2, - 7/2)
To solve 2x2+ 4x + 1 = 0
=> y = 0
When y = 0
x = -2 + √ 7/2 or x = -2 - √ 7/2
